Lemma 3.6 Let \(B\)  be a \((\alpha ,\beta )\) -approximation of \(P\) , and let \(C\)  be a \((\gamma ,\lambda )\) -approximation of \(B\) . Then \(C\)  is a \((\rho \alpha + 2 \rho ^2 \gamma (\alpha + 1),\lambda )\) -approximation of \(A\) .

Let \(\pi : P \rightarrow B\)  be the \((\alpha ,\beta )\) -approximation of \(P\)  and let \(t : B \rightarrow C\)  be the \((\gamma ,\lambda )\) -approximation of \(B\) .
In the following, all sums will be taken over all \(p \in P\) .
The hypotheses state that \(\sum D(p,\pi (p)) \le \alpha \text{\footnotesize \textsf {OPT}}(P)\)  and \(\sum D(\pi (p),t(\pi (p))) \le \gamma \text{\footnotesize \textsf {OPT}}(B)\) .
Let \(P^*\)  be an optimal clustering of \(P\) , that is \(\sum D(p,P^*) = \text{\footnotesize \textsf {OPT}}(P)\) .
Then \(\frac{1}{2} \text{\footnotesize \textsf {OPT}}(B) \le \sum D(\pi (p),P^*) \le \rho \sum (D(\pi (p),p) + D(p,P^*)) \le \rho (\alpha + 1) \text{\footnotesize \textsf {OPT}}(P)\) . The factor of \(\frac{1}{2}\)  comes from the fact that \(\text{\footnotesize \textsf {OPT}}(B)\)  is defined using centers restricted to \(B\)  (see [1]} for details).
We now write \(\sum D(p, t(\pi (p))) \le \rho \sum (D(p,\pi (p)) + D(\pi (p),t(\pi (p)))) \le (\rho \alpha + 2 \rho ^2 \gamma (\alpha + 1)) \text{\footnotesize \textsf {OPT}}(P)\)  as desired.

